Theorem infdifsn 9578

Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)

Assertion

Ref	Expression
infdifsn	⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem infdifsn

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		brdomi 8908	. . . 4 ⊢ (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1→𝐴)
2	1	adantr 480	. . 3 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → ∃𝑓 𝑓:ω–1-1→𝐴)
3		reldom 8901	. . . . . . 7 ⊢ Rel ≼
4	3	brrelex2i 5689	. . . . . 6 ⊢ (ω ≼ 𝐴 → 𝐴 ∈ V)
5	4	ad2antrr 727	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝐴 ∈ V)
6		simplr 769	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝐵 ∈ 𝐴)
7		f1f 6738	. . . . . . 7 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω⟶𝐴)
8	7	adantl 481	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω⟶𝐴)
9		peano1 7841	. . . . . 6 ⊢ ∅ ∈ ω
10		ffvelcdm 7035	. . . . . 6 ⊢ ((𝑓:ω⟶𝐴 ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ 𝐴)
11	8, 9, 10	sylancl 587	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓‘∅) ∈ 𝐴)
12		difsnen 8999	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ 𝐴 ∧ (𝑓‘∅) ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
13	5, 6, 11, 12	syl3anc 1374	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
14		vex 3446	. . . . . . . . . 10 ⊢ 𝑓 ∈ V
15		f1f1orn 6793	. . . . . . . . . . 11 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓:ω–1-1-onto→ran 𝑓)
16	15	adantl 481	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω–1-1-onto→ran 𝑓)
17		f1oen3g 8915	. . . . . . . . . 10 ⊢ ((𝑓 ∈ V ∧ 𝑓:ω–1-1-onto→ran 𝑓) → ω ≈ ran 𝑓)
18	14, 16, 17	sylancr 588	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ≈ ran 𝑓)
19	18	ensymd 8954	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ≈ ω)
20	3	brrelex1i 5688	. . . . . . . . . . 11 ⊢ (ω ≼ 𝐴 → ω ∈ V)
21	20	ad2antrr 727	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ∈ V)
22		limom 7834	. . . . . . . . . . 11 ⊢ Lim ω
23	22	limenpsi 9092	. . . . . . . . . 10 ⊢ (ω ∈ V → ω ≈ (ω ∖ {∅}))
24	21, 23	syl 17	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ≈ (ω ∖ {∅}))
25	14	resex 5996	. . . . . . . . . . 11 ⊢ (𝑓 ↾ (ω ∖ {∅})) ∈ V
26		simpr 484	. . . . . . . . . . . 12 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓:ω–1-1→𝐴)
27		difss 4090	. . . . . . . . . . . 12 ⊢ (ω ∖ {∅}) ⊆ ω
28		f1ores 6796	. . . . . . . . . . . 12 ⊢ ((𝑓:ω–1-1→𝐴 ∧ (ω ∖ {∅}) ⊆ ω) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
29	26, 27, 28	sylancl 587	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
30		f1oen3g 8915	. . . . . . . . . . 11 ⊢ (((𝑓 ↾ (ω ∖ {∅})) ∈ V ∧ (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅}))) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
31	25, 29, 30	sylancr 588	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
32		f1orn 6792	. . . . . . . . . . . . 13 ⊢ (𝑓:ω–1-1-onto→ran 𝑓 ↔ (𝑓 Fn ω ∧ Fun ◡𝑓))
33	32	simprbi 497	. . . . . . . . . . . 12 ⊢ (𝑓:ω–1-1-onto→ran 𝑓 → Fun ◡𝑓)
34		imadif 6584	. . . . . . . . . . . 12 ⊢ (Fun ◡𝑓 → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
35	16, 33, 34	3syl 18	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
36		f1fn 6739	. . . . . . . . . . . . . 14 ⊢ (𝑓:ω–1-1→𝐴 → 𝑓 Fn ω)
37	36	adantl 481	. . . . . . . . . . . . 13 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝑓 Fn ω)
38		fnima 6630	. . . . . . . . . . . . 13 ⊢ (𝑓 Fn ω → (𝑓 “ ω) = ran 𝑓)
39	37, 38	syl 17	. . . . . . . . . . . 12 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ ω) = ran 𝑓)
40		fnsnfv 6921	. . . . . . . . . . . . . 14 ⊢ ((𝑓 Fn ω ∧ ∅ ∈ ω) → {(𝑓‘∅)} = (𝑓 “ {∅}))
41	37, 9, 40	sylancl 587	. . . . . . . . . . . . 13 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → {(𝑓‘∅)} = (𝑓 “ {∅}))
42	41	eqcomd 2743	. . . . . . . . . . . 12 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ {∅}) = {(𝑓‘∅)})
43	39, 42	difeq12d 4081	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝑓 “ ω) ∖ (𝑓 “ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
44	35, 43	eqtrd 2772	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓 “ (ω ∖ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
45	31, 44	breqtrd 5126	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
46		entr 8955	. . . . . . . . 9 ⊢ ((ω ≈ (ω ∖ {∅}) ∧ (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
47	24, 45, 46	syl2anc 585	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
48		entr 8955	. . . . . . . 8 ⊢ ((ran 𝑓 ≈ ω ∧ ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
49	19, 47, 48	syl2anc 585	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
50		difexg 5276	. . . . . . . 8 ⊢ (𝐴 ∈ V → (𝐴 ∖ ran 𝑓) ∈ V)
51		enrefg 8933	. . . . . . . 8 ⊢ ((𝐴 ∖ ran 𝑓) ∈ V → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
52	5, 50, 51	3syl 18	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
53		disjdif 4426	. . . . . . . 8 ⊢ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅
54	53	a1i 11	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅)
55		difss 4090	. . . . . . . . . 10 ⊢ (ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓
56		ssrin 4196	. . . . . . . . . 10 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓 → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)))
57	55, 56	ax-mp 5	. . . . . . . . 9 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓))
58		sseq0 4357	. . . . . . . . 9 ⊢ ((((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
59	57, 53, 58	mp2an 693	. . . . . . . 8 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅
60	59	a1i 11	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
61		unen 8994	. . . . . . 7 ⊢ (((ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓)) ∧ ((ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅ ∧ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
62	49, 52, 54, 60, 61	syl22anc 839	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
63	8	frnd 6678	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ran 𝑓 ⊆ 𝐴)
64		undif 4436	. . . . . . 7 ⊢ (ran 𝑓 ⊆ 𝐴 ↔ (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
65	63, 64	sylib 218	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
66		uncom 4112	. . . . . . 7 ⊢ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)}))
67		eldifn 4086	. . . . . . . . . . 11 ⊢ ((𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓) → ¬ (𝑓‘∅) ∈ ran 𝑓)
68		fnfvelrn 7034	. . . . . . . . . . . 12 ⊢ ((𝑓 Fn ω ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ ran 𝑓)
69	37, 9, 68	sylancl 587	. . . . . . . . . . 11 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝑓‘∅) ∈ ran 𝑓)
70	67, 69	nsyl3 138	. . . . . . . . . 10 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
71		disjsn 4670	. . . . . . . . . 10 ⊢ (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ ↔ ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
72	70, 71	sylibr 234	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅)
73		undif4 4421	. . . . . . . . 9 ⊢ (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
74	72, 73	syl 17	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
75		uncom 4112	. . . . . . . . . 10 ⊢ ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓))
76	75, 65	eqtrid 2784	. . . . . . . . 9 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = 𝐴)
77	76	difeq1d 4079	. . . . . . . 8 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}) = (𝐴 ∖ {(𝑓‘∅)}))
78	74, 77	eqtrd 2772	. . . . . . 7 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (𝐴 ∖ {(𝑓‘∅)}))
79	66, 78	eqtrid 2784	. . . . . 6 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = (𝐴 ∖ {(𝑓‘∅)}))
80	62, 65, 79	3brtr3d 5131	. . . . 5 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → 𝐴 ≈ (𝐴 ∖ {(𝑓‘∅)}))
81	80	ensymd 8954	. . . 4 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴)
82		entr 8955	. . . 4 ⊢ (((𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
83	13, 81, 82	syl2anc 585	. . 3 ⊢ (((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝑓:ω–1-1→𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
84	2, 83	exlimddv 1937	. 2 ⊢ ((ω ≼ 𝐴 ∧ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
85		difsn 4756	. . . 4 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∖ {𝐵}) = 𝐴)
86	85	adantl 481	. . 3 ⊢ ((ω ≼ 𝐴 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) = 𝐴)
87		enrefg 8933	. . . . 5 ⊢ (𝐴 ∈ V → 𝐴 ≈ 𝐴)
88	4, 87	syl 17	. . . 4 ⊢ (ω ≼ 𝐴 → 𝐴 ≈ 𝐴)
89	88	adantr 480	. . 3 ⊢ ((ω ≼ 𝐴 ∧ ¬ 𝐵 ∈ 𝐴) → 𝐴 ≈ 𝐴)
90	86, 89	eqbrtrd 5122	. 2 ⊢ ((ω ≼ 𝐴 ∧ ¬ 𝐵 ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
91	84, 90	pm2.61dan 813	1 ⊢ (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ∖ cdif 3900   ∪ cun 3901   ∩ cin 3902   ⊆ wss 3903  ∅c0 4287  {csn 4582   class class class wbr 5100  ^◡ccnv 5631  ran crn 5633   ↾ cres 5634   “ cima 5635  Fun wfun 6494   Fn wfn 6495  ⟶wf 6496  –1-1→wf1 6497  –1-1-onto→wf1o 6499  ‘cfv 6500  ωcom 7818   ≈ cen 8892   ≼ cdom 8893

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5527  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-om 7819  df-er 8645  df-en 8896  df-dom 8897

This theorem is referenced by:  infdiffi  9579  infdju1  10112  infpss  10138